Collaborative Research: Midwest Geometry Conference 2005-2007

获奖：DMS 0509068, 0508661, 0509714首席研究员：Thomas P. Branson, Shihshu Walter Wei, Andrzej J. derdzinsky机构：爱荷华大学，俄克拉荷马大学，俄亥俄州立大学中西部几何会议自1991年以来每年召开一次。它的目标是向来自中西部的几何学者，包括初级研究人员和一些研究生介绍微分几何、几何分析和积分几何的最新进展。该计划的主要部分包括由特定子领域（不一定是中西部地区）的专家进行的几次全体讲座；此外，部分参加者亦有机会作专题演讲。当前的项目提供了2005年、2006年和2007年春季在俄亥俄州立大学、俄克拉荷马大学和爱荷华大学举行的会议。俄亥俄州会议计划于2005年4月29日至5月1日举行，将集中讨论三个主题：正弯曲流形；里奇流、孤子和爱因斯坦度规；几何群论。其他两个会议的主题列表仍是暂定的，包括：对称临界性；p-harmonic几何;凸性、正弯曲流形和Ricci流（Oklahoma, 2006），以及：偏微分方程的对称性和超定系统；theAdS /钢管对应;非线性双曲型演化方程；几何散射理论（爱荷华州，2007）。中西部几何会议致力于中西部几何社区成员之间的思想交流，以及在全国和世界范围内传播重要的研究进展和思想，在微分几何和相关领域。会议的一个主要目标是试图鼓励中西部地区的研究人员和潜在的研究人员，特别是那些工作地点离大型研究中心较远的研究人员，以及年轻人、妇女和代表性不足的群体的成员。为研究生和年轻研究人员准备的“热身”会议计划在会议前几天举行。每次会议的三个或四个焦点中至少有一个与几何应用有关。从历史上看，这种应用主要存在于物理学中，爱因斯坦的广义相对论和目前公认的基本粒子及其相互作用的标准模型提供了突出的例子。微分几何的其他著名应用是在工程上，例如机器人。另一个例子与进化方程有关，这是微分几何学者目前非常感兴趣的一个主题，同时，它在描述某些材料或空间本身如何随时间发展方面具有应用价值。后一类的一个非常具体的例子是平均曲率流。熔化的塑料球的表面随着时间的推移演变或适应，具有最小可能的曲率；这种演化完全由球体的曲率决定。这一基本观察提供了对各种流动问题的理解，并提供了一种预测流动的方法。这与材料的注塑成型直接相关，例如汽车的仪表板和保险杠。但是平均曲率流背后的基本原理也为像里奇流这样的问题提供了一个范例，在里奇流中，空间本身的度量特性是不断发展的量。后一类问题将是即将召开的中西部几何会议的中心议题。

AbstractAward: DMS 0509068, 0508661, 0509714Principal Investigator: Thomas P. Branson, Shihshu Walter Wei, Andrzej J. DerdzinskiInstitutions: University of Iowa, University of Oklahoma,Ohio State UniversityThe Midwest Geometry Conference has been meeting annually since1991. Its goal is to present recent advances in differentialgeometry, geometric analysis, and integral geometry to geometersfrom the Midwest, including junior level researchers and somegraduate students. The main part of the program consists ofseveral plenary lectures given by experts in specific sub-fields(who need not be Midwest-based); an opportunity is also providedfor some of the participants to give contributed talks. Thecurrent project provides for conferences in the spring of 2005,2006, and 2007 at the Ohio State University, the University ofOklahoma and, respectively, the University of Iowa. The OhioState conference, planned for April 29 to May 1, 2005, will focuson three topics: positively curved manifolds; Ricci flow,solitons and Einstein metrics; geometric group theory. The listsof topics for the other two conferences are still tentative, andinclude: symmetric criticality; p-harmonic geometry; convexity,positively curved manifolds, and Ricci flow (Oklahoma, 2006), aswell as: symmetries and overdetermined systems of PDEs; theAdS/CFT correspondence; nonlinear hyperbolic evolution equations;geometric scattering theory (Iowa, 2007).The Midwest Geometry Conference is dedicated to the exchange ofideas among members of the geometry community in the Midwest, aswell as dissemination of important research advances and ideas,nationwide and worldwide, in differential geometry and relatedfields. A key objective of the conference is the attempt toencourage researchers and potential researchers in the Midwest,especially those who work some distance from the large researchcenters, as well as young people, women, and members ofunderrepresented groups. "Warm-up" sessions for graduatestudents and young researchers are planned for the days leadingup to the conference. At least one of the three or four focustopics at each conference is related to applications of geometry.Historically, such applications lie primarily in physics, withprominent examples provided by Einstein's general theory ofrelativity and the currently accepted Standard Model ofelementary particles and their interactions. Other well-knownapplications of differential geometry are in engineering, forexample robotics. A further example pertains to evolutionequations, which are currently a subject of keen interest todifferential geometers and, at the same time, have the appliedaspect of describing how some materials, or the space itself,develop in time. A very specific example of the latter kind isprovided by the mean curvature flow. The surface of a globule ofmolten plastic evolves, or adapts in time, to have the minimumpossible curvature; this evolution is completely determined bythe globule's curvature. This fundamental observation providesan understanding of a wide range of flow problems, and allows anapproach to predicting the flow. This has immediate relevance toinjection molding of materials, for example in the dashboards andbumpers of automobiles. But the fundamental principles behindmean curvature flow have also provided a paradigm for problemslike the Ricci flow, in which the metric properties of spaceitself are the quantities which are evolving. Questions of thelatter kind will be central to the coming meetings of the MidwestGeometry Conference.